Probabilistic Implicational Bases in FCA and Probabilistic Bases of GCIs in EL⊥

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and Probabilistic Bases of GCIs in EL

^⊥

Francesco Kriegel

Institute for Theoretical Computer Science, TU Dresden, Germany [email protected]

http://lat.inf.tu-dresden.de/˜francesco

Abstract. A probabilistic formal context is a triadic context whose third dimension is a set of worlds equipped with a probability measure. After a formal definition of this notion, this document introduces probability of implications, and provides a construction for a base of implications whose probability satisfy a given lower threshold. A comparison between confidence and probability of implications is drawn, which yields the fact that both measures do not coincide, and cannot be compared. Furthermore, the results are extended towards the light-weight description logicEL^⊥with probabilistic interpretations, and a method for computing a base of general concept inclusions whose probability fulfill a certain lower bound is proposed.

Keywords: Formal Concept Analysis, Description Logics, Probabilistic Formal Context, Probabilistic Interpretation, Implication, General Concept Inclusion

1 Introduction

Most data-sets from real-world applications contain errors and noise. Hence, for mining them special techniques are necessary in order to circumvent the expression of the errors. This document focuses on rule mining, especially we attempt to extract rules that are approximately valid in data-sets, or families of data-sets, respectively. There are at least two measures for the approximate soundness of rules, namelyconfidence andprobability. While confidence expresses the number of counterexamples in a single data-set, probability expresses somehow the number of data-sets in a data-set family that do not contain any counterexample. More specifically, we consider implications in the formal concept analysis setting [7], and general concept inclusions (GCIs) in the description logics setting [1] (in the light-weight description logicEL^⊥^).

Firstly, for axiomatizing rules from formal contexts possibly containing wrong incidences or having missing incidences the notion of a partial implication (also called association rule) and confidence has been defined by Luxenburger in [12]. Further- more, Luxenburger introduced a method for the computation of a base of all partial implications holding in a formal context whose confidence is above a certain threshold.

In [2] Borchmann has extended the results to the description logicEL^⊥by adjusting the notion of confidence to GCIs, and also gave a method for the construction of a base of confident GCIs for an interpretation.

Secondly, another perspective is a family of data-sets representing different views of the same domain, e.g., knowledge of different persons, or observations of an exper- iment that has been repeated several times, since some effects could not be observed

c paper author(s), 2015. Published in Sadok Ben Yahia, Jan Konecny (Eds.): CLA 2015, pp. 193–204, ISBN 978–2–9544948–0–7, Blaise Pascal University, LIMOS laboratory, Clermont-Ferrand, 2015. Copying permitted only for private and academic purposes.

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in every case. In the field of formal concept analysis, Vityaev, Demin, and Ponomaryov, have introduced in probabilistic extensions of formal contexts and their formal concepts and implications, and furthermore gave some methods for their computation, cf. [4].

In [9] the author has shown some methods for the computation of a base of GCIs in probabilistic description logics where concept and role constructors are available to express probability directly in the concept descriptions. Here, we want to use another approach, and do not allow for probabilistic constructors, but define the notion of a probability of general concept inclusions in the light-weight description logicEL^⊥^. Furthermore, we provide a method for the computation of a base of GCIs satisfying a certain lower threshold for the probability. More specifically, we use the description logicEL^⊥with probabilistic interpretations that have been introduced by Lutz and Schröder in [11]. Beforehand, we only consider conjunctions in the language of formal concept analysis, and define the notion of a probabilistic formal context in a more general form than in [4], and provide a technique for the computation of base of implications satisfying a given lower probability threshold.

The document is structured as follows. In Section 2 some basic notions for probabilistic extensions of formal concept analysis are defined. Then, in Section 3 a method for the computation of a base for all implications satisfying a given lower probability threshold in a probabilistic formal context is developed, and its correctness is proven. The following sections extend the results to the description logicEL^⊥. In particular, Section 4 introduces the basic notions forEL^⊥, and defines probabilistic interpretations. Section 5 shows a technique for the construction of a base of GCIs holding in a probabilistic interpretation and fulfilling a lower probability threshold. Furthermore, a comparison of the notions of confidence and probability is drawn at the end of Section 3.

2 Probabilistic Formal Concept Analysis

Aprobability measurePon a countable setWis a mappingP: 2^W→[0, 1]such that P(∅) = 0 andP(W) = 1 hold, andPis σ-additive, i.e., for all pairwise disjoint countable families(Un)_n∈NwithUn⊆Wit holds thatP(^S_n∈NUn) =∑_n∈NP(Un). A worldw∈WispossibleifP{w}>0 holds, andimpossibleotherwise. The set of all possible worlds is denoted byWε, and the set of all impossible worlds is denoted byW0. Obviously,Wε]W0is a partition ofW.

Definition 1 (Probabilistic Formal Context).Aprobabilistic formal contextKis a tuple(G,M,W,I,P)that consists of a set G ofobjects, a set M ofattributes, a countable set W of worlds, anincidence relationI⊆G×M×W, and a probability measurePon W.

For a triple(g,m,w)∈I we say thatobjectghas attributemin worldw. Furthermore, we define thederivations in worldw as operators·^I^w^{: 2}^G→²^Mand·^I^w^{: 2}^M→²^Gwhere

A^I^w :={m∈M| ∀g∈ A:(g,m,w)∈I} B^I^w :={g∈G| ∀m∈B:(g,m,w)∈I}

for object sets A⊆G and attribute sets B⊆M, i.e., A^I^wis the set of all common attributes of all objects in A in the world w, and B^I^w is the set of all objects that have all attributes in B in w.

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Definition 2 (Implication, Probability).Let K = (G,M,W,I,P)be a probabilistic formal context. For attribute sets X,Y⊆M we call X→Y animplicationover M, and its probability inKis defined as the measure of the set of worlds it holds in, i.e.,

P(X→Y):=P{w∈W|X^I^w ⊆Y^I^w}^. Furthermore, we define the following properties for implications X→Y:

1. X→Yholdsin world w ofKif X^I^w ⊆Y^I^w is satisfied.

2. X→Ycertainly holdsinKif it holds in all worlds ofK.

3. X→Yalmost certainly holdsinKif it holds in all possible worlds ofK.

4. X→Ypossibly holdsinKif it holds in a possible world ofK. 5. X→Y isimpossibleinKif it does not hold in any possible world ofK. 6. X→Y isrefutedbyKif does not hold in any world ofK.

It is readily verified thatP(X→Y) =P{w∈Wε|XÎ^w ⊆YÎ^w}=∑{^P{w} |w∈ WεandXÎ^w ⊆YÎ^w}. An implicationX→Yalmost certainly holds ifP(X→Y) =1, possibly holds ifP(X →Y) >0, and is impossible ifP(X →Y) = 0. IfX →Y certainly holds, then it is almost certain, and ifX→Yis refuted, then it is impossible.

3 Probabilistic Implicational Bases

At first we introduce the notion of a probabilistic implicational base. Then we will develop and prove a construction for such bases w.r.t. probabilistic formal contexts. If the underlying context is finite, then the base is computable. The reader should be aware of the standard notions of formal concept analysis in [7]. Recall that an implication follows from an implication set if, and only if, it can be syntactically deduced using the so-calledArmstrong rulesas follows: 1. FromX⊇YinferX→Y. 2. FromX→Y andY→ZinferX→Z. 3. FromX1→Y1andX2→Y2inferX1∪X2→Y1∪Y2. Definition 3 (Probabilistic Implicational Base).LetK= (G,M,W,I,P)be a probabilistic formal context, and p∈[0, 1]a threshold. Aprobabilistic implicational baseforK and p is an implication setBover M that satisfies the following properties:

1. BissoundforKand p, i.e.,P(X→Y)≥p holds for all implications X→Y∈ B, and 2. BiscompleteforKand p, i.e., ifP(X→Y)≥p, then X→Y follows fromB. A probabilistic implicational base isirredundantif none of its implications follows from the others, and isminimalif it has minimal cardinality among all bases forKand p.

It is readily verified that the above definition is a straight-forward generalization of implicational bases as defined in [7, Definition 37], in particular formal contexts coincide with probabilistic formal contexts having only one possible world, and implications holding in the formal context coincide with implications having probability 1.

We now define a transformation from probabilistic formal contexts to formal contexts.

It allows to decide whether an implication (almost) certainly holds, and furthermore it can be utilized to construct an implicational base for the (almost) certain implications.

Definition 4 (Scaling).LetKbe a probabilistic formal context. Thecertain scalingofKis the formal contextK^×:= (G×W,M,I^×)where((g,w),m)∈I^×iff(g,m,w)∈I, and thealmost certain scalingofKis the subcontextK^×_ε := (G×Wε,M,Iε^×)ofK^×.

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Lemma 5. LetK= (G,M,W,I,P)be a probabilistic formal context, and let X→Y be a formal implication. Then the following statements are satisfied:

1. X→Y certainly holds inKif, and only if, X→Y holds inK^×. 2. X→Y almost certainly holds inKif, and only if, X→Y holds inK^×_ε . Proof. It is readily verified that the following equivalences hold:

P(X→Y) =1⇔ ∀w∈W:X^I^w ⊆Y^I^w

⇔XÎ^×=^]_w∈WXÎ^w× {w} ⊆^]_w∈WYÎ^w× {w}=YÎ^×

⇔^K^×|=X→Y.

The second statement can be proven analogously. ut

Recall the notion of a pseudo-intent [6–8]: An attribute setP⊆Mof a formal context (G,M,I)is apseudo-intentifP6=P^II, andQ^II⊆Pholds for all pseudo-intentsQ(^P.

Furthermore, it is well-known that thecanonical implicational baseof a formal context (G,M,I)consists of all implicationsP→ P^IIwherePis a pseudo-intent, cf. [6–8].

Consequently, the next corollary is an immediate consequence of Lemma 5.

Corollary 6. LetKbe a probabilistic formal context. Then the following statements hold:

1. An implicational base forK^×is an implicational base for the certain implications ofK, in particular this holds for the following implication set:

BK:={P→P^I^×^I^×|P∈^PsInt(K^×)}^.

2. An implicational base forK^×_ε w.r.t. the background knowledgeBKis an implicational base for the almost certain implications ofK, in particular this holds for the following implication set:

BK,1:=BK∪ {P→P^I^ε^×^I^ε^×|P∈^PsInt(K^×_ε ,BK)}^.

Lemma 7. LetK= (G,M,W,I,P)be a probabilistic formal context. Then the following statements are satisfied:

1. Y⊆X implies that X→Y certainly holds inK.

2. X1⊆X2and Y1⊇Y2implyP(X1→Y1)≤^P(X2→Y2).

3. X0⊆X1⊆^{. . .}⊆XnimpliesP(X0→Xn)≤^Vⁿ_i=1^P(Xi−1→Xi). Proof. 1. IfY⊆X, thenX^I^w ⊆Y^I^wfollows for all worldsw∈W.

2. AssumeX1⊆ X2andY2 ⊆Y1. ThenX₁Î^w ⊇X₂Î^w andY₂Î^w ⊇Y₁Î^w follow for all worldsw∈W. Consider a worldw∈WwhereX₁Î^w ⊆Y₁Î^w. Of course, we may conclude thatX₂Î^w ⊆Y₂Î^w. As a consequence we getP(X1→Y1)≤^P(X2→Y2). 3. We prove the third claim by induction onn. Forn=0 there is nothing to show, and the case n = 1 is trivial. Hence, consider n = 2 for the induction base, and letX0 ⊆ X1 ⊆ X2. Then we have thatX₀Î^w ⊇ X₁Î^w ⊇ X₂Î^w is satisfied in all worldsw ∈ W. Now consider a worldw ∈ WwhereX₀Î^w ⊆ X₂Î^w is true.

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Of course, it then follows thatX₀Î^w ⊆ X₁Î^w ⊆ X₂Î^w. Consequently, we conclude P(X0→X2)≤^P(X0→X₁)andP(X0→X2)≤^P(X₁→X2).

For the induction step letn>2. The induction hypothesis yields that P(X0→Xn−1)≤^{^}ⁿ_i=1⁻¹^P(Xi−1→Xi).

Of course, it also holds thatX0 ⊆ Xn−1 ⊆ Xn, and it follows by induction hypothesis and the previous inequality that

P(X0→Xn)≤^P(X0→Xn−1)∧^P(Xn−1→Xn)≤^{^}ⁿ_i=1^P(Xi−1→Xi).ut Lemma 8. LetK= (G,M,W,I,P)be a probabilistic formal context. Then for all implications X→Y the following equalities are valid:

P(X→Y) =P(XÎ^×Î^× →YÎ^×Î^×) =P(XÎ^×^ε Î^ε^× →YÎ^×^ε Î^×^ε ). Proof. LetX→Ybe an implication. Then for all worldsw∈Wit holds that

g∈XÎ^w ⇔ ∀m∈X:(g,m,w)∈I⇔ ∀m∈X:((g,w),m)∈I^×⇔(g,w)∈XÎ^×, and we conclude thatXÎ^w = π₁(XÎ^×∩(G× {w})). Furthermore, we then infer XÎ^w =XÎ^×Î^×Î^w, and thus the following equations hold:

P(X→Y) =P{w∈W|X^I^w ⊆Y^I^w}

=P{w∈W|XÎ^×Î^×Î^w ⊆YÎ^×Î^×Î^w}=P(XÎ^×Î^× →YÎ^×Î^×). In particular, for all possible worldsw∈Wεit holds thatg∈XÎ^w ⇔(g,w)∈XÎ^ε^×, and thusXÎ^w =π₁(XÎ^×^ε ∩(G× {w}))ândXÎ^w =XÎ^ε^×Î^ε^×Î^ware satisfied. Consequently, it may be concluded thatP(X→Y) =P(XÎ^ε^×Î^ε^× →YÎ^ε^×Î^ε^×). ut Lemma 9. LetKbe a probabilistic formal context. Then the following statements hold:

1. IfBis an implicational base for the certain implications ofK, then the implication X→Y follows fromB ∪ {XÎ^×Î^× →YÎ^×Î^×}.

2. IfBis an implicational base for the almost certain implications ofK, then the implication X→Y follows fromB ∪ {XÎ^×^ε Î^ε^× →YÎ^ε^×Î^×^ε }.

Proof. Of course, the implicationX→XÎ^×Î^×holds inK^×, i.e., certainly holds inK by Lemma 5, and hence follows fromB. Thus, the implicationX→YÎ^×Î^× is entailed byB ∪ {XÎ^×Î^× →YÎ^×Î^×}, and because ofY⊆YÎ^×Î^×the claim follows.

The second statement follows analogously. ut

Lemma 10. LetKbe a probabilistic formal context. Then the following statements hold:

1. P(XÎ^×^ε Î^×^ε →YÎ^ε^×Î^×^ε ) =P(XÎ^ε^×Î^ε^×→(X∪Y)Î^ε^×Î^×^ε ), 2. (X∪Y)Î^ε^×Î^ε^× →YÎ^ε^×Î^ε^×certainly holds inK, and

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3. XÎ^ε^×Î^×^ε →YÎ^ε^×Î^ε^×is entailed by{XÎ^ε^×Î^ε^×→(X∪Y)Î^ε^×Î^×^ε ,(X∪Y)Î^×^ε Î^ε^× →YÎ^ε^×Î^×^ε }. (X∪Y)Î^ε^×Î^ε^×

XÎ^ε^×Î^×^ε p YÎ^ε^×Î^ε^× p

Proof. First note that(XÎ^ε^×Î^ε^×∪YÎ^ε^×Î^ε^×)Î^ε^×Î^×^ε = (X∪Y)Î^×^ε Î^ε^×. AsYÎ^ε^×Î^×^ε is a subset of (XÎ^ε^×Î^ε^×∪YÎ^×^ε Î^×^ε )Î^ε^×Î^ε^×, the implication(X∪Y)Î^ε^×Î^×^ε →YÎ^×^ε Î^ε^×certainly holds inK, cf.

Statement 1 in Lemma 7.

Furthermore, we have thatXÎ^w ⊆YÎ^wif, and only if,XÎ^w ⊆XÎ^w∩YÎ^w = (X∪Y)Î^w. Hence, the implicationX→Yhas the same probability asX→X∪Y. Consequently, we may conclude by means of Lemma 7 that

P(XÎ^ε^×Î^ε^×→YÎ^ε^×Î^ε^×) =P(X→Y) =P(X→X∪Y) =P(XÎ^×^ε Î^×^ε →(X∪Y)Î^×^ε Î^ε^×). Obviously,{XÎ^×^ε Î^ε^× →(X∪Y)Î^×^ε Î^×^ε ,(X∪Y)Î^×^ε Î^×^ε →YÎ^ε^×Î^ε^×}êntailsXÎ^ε^×Î^ε^× →YÎ^ε^×Î^ε^×.ut Lemma 11. LetKbe a probabilistic formal context, and X,Y be intents ofK^×_ε such that X⊆Y andP(X→Y)≥p. Then the following statements are true:

1. There is a chain of neighboring intents X=X0≺X1≺X2≺^{. . .}≺Xn =Y inK^×_ε , 2. P(Xi−1→Xi)≥ p for all i∈ {^{1, . . . ,}n}, and

3. X→Y is entailed by{X_i−1→X_i|i∈ {^{1, . . . ,}n} }.

Proof. The existence of a chainX = X0 ≺ X1 ≺ X2 ≺ ^{. . .}≺ Xn−1 ≺ Xn = Yof neighboring intents betweenXandYinK^×_ε follows fromX⊆Y.

From Statement 3 in Lemma 7 it follows that all implicationsX_i−1 →X_ihave a probability of at leastpinK. It is trivial that they entailX→Y. ut Theorem 12 (Probabilistic Implicational Base).LetKbe a probabilistic formal context, and p∈ [0, 1)a probability threshold. Then the following implication set is a probabilistic implicational base forKand p:

B^K,p:=BK,1∪ {X→Y|X,Y∈^Int(K^×_ε )and X≺Y andP(X→Y)≥p}^. Proof. All implications inBK,1hold almost certainly inK, and thus have probability 1.

By construction, all other implicationsX→Yin the second subset have a probability

≥p. Hence, Statement 1 in Definition 3 is satisfied.

Now consider an implicationX→YoverMsuch thatP(X→Y)≥ p. We have to prove Statement 2 of Definition 3, i.e., thatX→Yis entailed byB^K,p.

Lemma 8 yields that both implicationsX→YandXÎ^ε^×Î^ε^× →YÎ^ε^×Î^ε^×have the same probability. Lemma 9 states thatX →Yfollows fromBK,1∪ {XÎ^ε^×Î^×^ε →YÎ^ε^×Î^ε^×}^. According to Lemma 10, the implicationXÎ^×^ε Î^ε^× →YÎ^ε^×Î^ε^× follows from{XÎ^×^ε Î^ε^× → (X∪Y)Î^ε^×Î^ε^×,(X∪Y)Î^ε^×Î^ε^× →YÎ^ε^×Î^ε^×}. Furthermore, it holds that

P(XÎ^×^ε Î^ε^× →(X∪Y)Î^ε^×Î^ε^×) =P(XÎ^×^ε Î^ε^× →YÎ^×^ε Î^×^ε ) =P(X→Y)≥p,

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and the second implication(X∪Y)Î^×^ε Î^ε^× →YÎ^ε^×Î^ε^×certainly holds, i.e., follows from B^K,1. Finally, Lemma 11 states that there is a chain of neighboring intents ofK^×_ε starting atXÎ^ε^×Î^ε^×and ending at(X∪Y)Î^ε^×Î^×^ε , i.e.,

XÎ^×^ε Î^ε^× =X₀Î^×^ε Î^×^ε ≺X₁Î^ε^×Î^×^ε ≺X₂Î^ε^×Î^ε^× ≺^{. . .}≺X_nÎ^ε^×Î^×^ε = (X∪Y)Î^ε^×Î^ε^×,

such that all implicationsX_i−1Î^ε^×Î^ε^× →X_iÎ^ε^×Î^×^ε have a probability≥p, and are thus con- tained inBK,p. Hence,BK,pentails the implicationX→Y. ut Corollary 13. LetKbe a probabilistic formal context. Then the following set is an implicational base for the possible implications ofK:

BK,ε:=BK,1∪ {X→Y|X,Y∈^Int(K^×_ε )and X≺Y andP(X→Y)>0}^. However, it is not possible to show irredundancy or minimality for the base of probabilistic implications given above in Theorem 12. Consider the probabilistic formal contextK= ({g1,g2}^,{m1,m2}^,{w1,w2}^,I,{ {^w1} 7→ ¹₂^,{w2} 7→ ¹₂})^whose incidence relationIis defined as follows:

w1 m1m2

g1 × ×

g2 ×

w2 m1m2

g1 × × g2 × ×

(G×W,{m2})

({(g1,w1),(g1,w2),(g2,w2)}^,{m1,m2}) The only pseudo-intent ofK^×is∅, and the concept lattice ofK^×is shown above.

Hence, we have the following probabilistic implicational base forp= ¹₂: B_K,¹₂ ={^∅→ {m2}^,{m2} → {m1,m2} }^.

However, the setB^:={^∅→ {m1,m2} }is also a probabilistic implicational base for Kand¹₂with less elements.

In order to compute a minimal base for the implications holding in a probabilistic formal context with a probability≥p, one can for example determine the above given probabilistic base, and minimize it by means of constructing the Duquenne-Guigues base of it. This either requires the transformation of the implication set into a formal context that has this implication set as an implicational base, or directly compute all pseudo- closures of the closure operator induced by the (probabilistic) implicational base.

Recall that the confidence of an implicationX→Yin a formal context(G,M,I) is defined asconf(X→Y):=(X∪Y)^I/X^I, cf. [12]. In general, there is no corre- spondence between the probability of an implication inKand its confidence inK^× orK^×_ε. To prove this we will provide two counterexamples. As first counterexample we consider the contextKabove. It is readily verified thatP({m2} → {m1}) = ¹₂ andconf({m2} → {m1}) = ³₄, i.e., the confidence is greater than the probability.

Furthermore, consider the following modification ofKas second counterexample:

w1 m1m2

g1 × × g2

w2 m1m2

g1 ×

g2 ×

Then we have thatP({m2} → {m₁}) = ¹₂^and^conf({m2} → {m₁}) = ¹₃, i.e., the confidence is smaller than the probability.

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4 The Description Logic EL

^⊥

and Probabilistic Interpretations

This section gives a brief overview on the light-weight description logicEL^⊥[1]. First, assume that(N_C,N_R)is a signature, i.e., N_Cis a set ofconcept names, andN_Ris a set ofrole names, respectively. ThenEL^⊥-concept descriptions Cover(NC,NR)may be constructed according to the following inductive rule (whereA∈NCandr∈NR):

C::=⊥ | > |A|CuC| ∃r.C.

We shall denote the set of allEL^⊥-concept descriptions over(NC,NR)byEL^⊥(NC,NR). Second, the semantics ofEL^⊥-concept descriptions is defined by means of interpretations: Aninterpretationis a tupleI = (∆Î,·Î)that consists of a set∆Î, calleddomain, and anextension function·Î^: NC∪NR →²^∆Î ∪²^∆Î^×^∆Î that maps concept names A∈NCto subsetsAÎ ⊆^∆Î and role namesr∈NRto binary relationsrÎ ⊆^∆Î×^∆Î^. The extension function is extended to allEL^⊥-concept descriptions as follows:

⊥^I ^:=∅,

>Î ^:=∆Î, (CuD)Î :=CÎ∩DÎ,

(∃r.C)Î :={d∈^∆Î| ∃e∈^∆Î^:(d,e)∈rÎande∈CÎ}^.

Ageneral concept inclusion (GCI)inEL^⊥is of the formCvDwhereCandDareEL^⊥^- concept descriptions. Itholdsin an interpretationIîfCÎ ⊆DÎis satisfied, and we then also writeI |=CvD, and say thatI^{is a}modelofCvD. Furthermore,Cissubsumed byDifCvDholds in all interpretations, and we shall denote this byCvD, too. A TBoxis a set of GCIs, and amodelof a TBox is a model of all its GCIs. A TBoxT entails a GCICvD, denoted byT |=CvD, if every model ofT is a model ofCvD.

To introduce probability into the description logicEL^⊥, we now present the notion of a probabilistic interpretation from [11]. It is simply a family of interpretations over the same domain and the same signature, indexed by a set of worlds that is equipped with a probability measure.

Definition 14 (Probabilistic Interpretation, [11]).Let(NC,NR)be a signature. Aprob- abilistic interpretationI is a tuple(∆Î,(·Î^w)_w_∈_W,W,P)consisting of a set∆Î, called domain, a countable set W of worlds, a probability measurePon W, and anextension function·Î^wfor each world w∈W, i.e.,(∆Î,·Î^w)is an interpretation for each w∈W.

For a general concept inclusion CvD itsprobabilityinI is defined as follows:

P(CvD):=P{w∈W|C^I^w ⊆D^I^w}^.

Furthermore, for a GCI CvD we define the following properties (as for probabilistic formal contexts): 1. CvDholdsin world w if C^I^w ⊆D^I^w. 2. CvDcertainly holdsinIif it holds in all worlds. 3. CvDalmost certainly holdsinIif it holds in all possible worlds.

4. CvDpossibly holdsinIif it holds in a possible world. 5. CvD isimpossiblein Iif it does not hold in any possible world. 6. CvD isrefutedbyIif it does not hold in any world.

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It is readily verified thatP(CvD) =P{w∈Wε|CÎ^w ⊆DÎ^w}=∑{^P{w} |w∈ WεandCÎ^w ⊆DÎ^w}for all general concept inclusionsCvD.

5 Probabilistic Bases of GCIs

In the following we construct from a probabilistic interpretationIa base of GCIs that entails all GCIs with a probability greater than a given thresholdpw.r.t.I^.

Definition 15 (Probabilistic Base).LetIbe a probabilistic interpretation, and p∈[0, 1] a threshold. Aprobabilistic base of GCIsforIand p is a TBoxBthat satisfies the following conditions:

1. BissoundforIand p, i.e.,P(CvD)≥p for all GCIs CvD∈ B, and 2. BiscompleteforIand p, i.e., ifP(CvD)≥p, thenB |=CvD.

A probabilistic base Bisirredundantif none of its GCIs follows from the others, and is minimalif it has minimal cardinality among all probabilistic bases forIand p.

For a probabilistic interpretationIwe define itscertain scalingas the disjoint union of all interpretationsIwwithw∈W, i.e., as the interpretationI^×^:= (∆^I×W,·^I^×) whose extension mapping is given as follows:

AÎ^× :={(d,w)|d∈ AÎ^w} (A∈NC), rÎ^× :={((d,w),(e,w))|(d,e)∈rÎ^w} (r∈NR).

Furthermore, thealmost certain scalingIε^×ofIis the disjoint union of all interpretations Iwwherew∈Wεis a possible world. Analogously to Lemma 5, a GCICvDcertainly holds inIiff it holds inI^×, and almost certainly holds inIiff it holds inIε^×.

In [5] the so-calledmodel-based most-specific concept descriptions (mmscs)have been defined w.r.t. greatest fixpoint semantics as follows: LetJbe an interpretation, andX⊆

∆^J. Then a concept descriptionCis ammscofXinJ^{, if}X⊆C^J is satisfied, andCv Dfor all concept descriptionsDwithX⊆D^J. It is easy to see that all mmscs ofXare unique up to equivalence, and hence we denotethemmsc ofXinJ^byX^J. Please note that there is also a role-depth bounded variant w.r.t. descriptive semantics given in [3].

Lemma 16. LetIbe a probabilistic interpretation. Then the following statements hold:

1. CÎ^w× {w}=CÎ^×∩(∆Î× {w})for all concept descriptions C and worlds w∈W.

2. CÎ^w× {w}=CÎ^ε^×∩(∆Î× {w})for all concept descriptions C and possible worlds w∈Wε.

3. P(CvD) =P(CÎ^×Î^× vDÎ^×Î^×) =P(CÎ^ε^×Î^×^ε vDÎ^ε^×Î^ε^×)for all GCIs CvD.

Proof. 1. We prove the claim by structural induction onC. By definition, the statement holds for⊥^,>, and all concept namesA∈ N_C. Consider a conjunctionCuD, then

(CuD)Î^w× {w}= (CÎ^w∩DÎ^w)× {w}

=C^I^w× {w} ∩D^I^w× {w}

I.H.=CÎ^×∩DÎ^×∩(∆Î× {w})

= (CuD)^I^×∩(∆^I× {w}).

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For an existential restriction∃r.Cthe following equalities hold:

(∃r.C)^I^w× {w}

={d∈^∆Î| ∃e∈^∆Î^:(d,e)∈rÎ^w ande∈CÎ^w} × {w}

={(d,w)| ∃(e,w):((d,w),(e,w))∈r^I^×and(e,w)∈C^I^w× {w} }

I.H.={(d,w)| ∃(e,w):((d,w),(e,w))∈r^I^×and(e,w)∈C^I^×}

= (∃r.C)^I^×∩(∆^I× {w}). 2. analogously.

3. Using the first statement we may conclude that the following equalities hold:

P(CvD)

=P{w∈W|C^I^w× {w} ⊆D^I^w× {w} }

=P{w∈W|CÎ^×∩(∆Î× {w})⊆DÎ^×∩(∆Î× {w})}

=P{w∈W|CÎ^×Î^×Î^×∩(∆Î× {w})⊆DÎ^×Î^×Î^×∩(∆Î× {w})}

=P{w∈W|CÎ^×Î^×Î^w× {w} ⊆DÎ^×Î^×Î^w× {w} }

=P(CÎ^×Î^× vDÎ^×Î^×).

The second equality follows analogously. ut

For a probabilistic interpretationI = (∆Î,·Î^,W,P)and a setMofEL^⊥^-concept descriptions we define theirinduced contextas the probabilistic formal contextK_I_,M:= (∆Î,M,W,I,P)where(d,C,w)∈Iiffd∈CÎ^w.

Lemma 17. LetI be a probabilistic interpretation, M a set ofEL^⊥-concept descriptions, and X,Y⊆M. Then the probability of the implication X→Y in the induced contextK_I_,Mequals the probability of the GCIdXv^dY inI, i.e., it holds thatP(X→Y) =P(^dXv^dY). Proof. The following equivalences are satisfied for allZ⊆Mand worldsw∈W:

d∈ZÎ^w ⇔ ∀C∈Z:(d,C,w)∈I⇔ ∀C∈Z:d∈CÎ^w ⇔d∈(^lZ)Î^w. Now consider two subsetsX,Y⊆M, then it holds that

P(X→Y) =P{w∈W|X^I^w ⊆Y^I^w}

=P{w∈W|(^lX)^I^w ⊆(^lY)^I^w}=P(^lXv^lY). ut Analogously to [5], the contextK_Iis defined asK_I_,M_I with the following attributes:

M_I :={ ⊥ } ∪NC∪ { ∃r.X^I^ε^×|^∅6=X⊆^∆^I×Wε}^.

For an implication setB^{over a set}MofEL^⊥-concept descriptions we define its induced TBoxbyd

B^:={^dXv^dY|X→Y∈ B }^.

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Corollary 18. IfBcontains an almost certain implicational base forK_I, thend

Bis complete for the almost certain GCIs ofI.

Proof. We know that a GCI almost certainly holds inI if, and only if, it holds in Iε^×. LetB⁰⊆ Bbe an almost certain implicational base forK_I, i.e., an implicational base for(K_I)^×_ε =K_I×

ε . Then according to Distel in [5, Theorem 5.12] it follows that the TBoxd

B⁰is a base of GCIs forIε^×, i.e., a base for the almost certain GCIs ofI^. Consequently,d

Bis complete for the almost certain GCIs ofI^. ut Theorem 19. LetI be a probabilistic interpretation, and p ∈ [0, 1]a threshold. IfBis a probabilistic implicational base forK_I and p that contains an almost certain implicational base forK_I, thend

Bis a probabilistic base of GCIs forIand p.

Proof. Consider a GCIdXv^dY∈^dB. Then Lemma 17 yields that the implication X→Yand the GCIdXv^dYhave the same probability. SinceBis a probabilistic implicational base forK_Iandp, we conclude thatP(^dXv^dY)≥pis satisfied.

Assume thatCv Dis an arbitrary GCI with probability≥ p. We have to show thatd

B ^entailsC v D. LetJ be an arbitrary model ofd

B. Consider an implication X → Y ∈ B^{, then}^dX v ^dY ∈ ^dB holds, and hence it follows that (^dX)^J ⊆(^dY)^J. Consequently, the implicationX→Yholds in the induced con- textK_J_,M_I. (We here mean the non-probabilistic formal context that is induced by a non-probabilistic interpretation, cf. [2, 3, 5].)

Furthermore, since all model-based most-specific concept descriptions ofIε^×are expressible in terms ofM_I, we have thatE≡^d^πM_I(E)holds for all mmscsEofIε^×, cf. [2, 3, 5]. Hence, we may conclude that

P(CvD) =P(CÎ^×^ε Î^ε^× vDÎ^×^εÎ^ε^×)

=P(^lπ_M_I(CÎ^ε^×Î^ε^×)v^l^πM_I(DÎ^ε^×Î^ε^×))

=P(π_M_I(CÎ^ε^×Î^ε^×)→^πM_I(DÎ^ε^×Î^ε^×)).

Consequently,Bentails the implicationπ_M_I(CÎ^ε^×Î^ε^×)→^πM_I(DÎ^ε^×Î^ε^×), hence it holds inK_J_,M_I, and furthermore the GCICÎ^ε^×Î^ε^× vDÎ^ε^×Î^ε^×holds inJ^{. As}J is an arbitrary interpretation,d

BêntailsCÎ^ε^×Î^ε^× vDÎ^ε^×Î^ε^×. Corollary 18 yields thatd

Bis complete for the almost certain GCIs ofI^{. In par-} ticular, the GCICvC^I^ε^×^I^ε^×almost certainly holds inI, and hence follows fromd

B^. We conclude thatd

B |=CvDÎ^ε^×Î^×^ε . Of course, the GCIDÎ^ε^×Î^ε^× v Dholds in all interpretations. Finally, we conclude thatd

B^entailsCvD. ut

Corollary 20. Let I be a probabilistic interpretation, and p ∈ [0, 1]a threshold. Then dB^K_I,pis a probabilistic base of GCIs forIand p whereB^K_I,pis defined as in Theorem 12.

6 Conclusion

We have introduced the notion of a probabilistic formal context as a triadic context whose third dimension is a set of worlds equipped with a probability measure. Then

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the probability of implications in such probabilistic formal contexts was defined, and a construction of a base of implications whose probability exceeds a given threshold has been proposed, and its correctness has been verified. Furthermore, the results have been applied to the light-weight description logicEL^⊥with probabilistic interpretations, and so we formulated a method for the computation of a base of general concept inclusions whose probability satisfies a given lower threshold.

For finite input data-sets all of the provided constructions are computable. In particular, [3, 5] provide methods for the computation of model-based most-specific concept descriptions, and the algorithms in [6, 10] can be utilized to compute concept lattices and canonical implicational bases (or bases of GCIs, respectively).

The author thanks Sebastian Rudolph for proof reading and a fruitful discussion, and the anonymous reviewers for their constructive comments.

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